We assume in our paper, as is common, that the data samples are independent and identically distributed. We note though that we took some care in our proofs to not exploit this fact explicitly. Indeed, we merely use that the empirical distribution \(\hat{\mathbb {P}}_T\)  constructed enjoys certain deviation principles (Theorem REF  and REF ).
Hence, our analysis should with some effort remain applicable in case the data is generated by any process for which \(\hat{\mathbb {P}}_T\)  verifies the appropriate deviation principles. We refer to [1]} for several examples of such processes.
